I have formulated something like
For $V_n$ where ${\bf x}=(x_1,x_2,x_3, \ldots,x_n)$, ${\bf y}=(y_1,y_2,y_3,\ldots,y_n)$
and I define ${\bf x} \cdot {\bf y}$ to be $(x_1+x_2+x_3+ \cdots+x_n)(y_1+y_2+y_3+\cdots+y_n)$
it seems like it works for all four axioms.
Is there mistake here?
$\def\\#1{{\bf#1}}$An inner product should have the property that if $\\x\cdot\\x=0$ then $\\x=\\0$. This is not true for your example (except in the case $n=1$). For example, if $\\x=(1,-1,0,0,\ldots)$, then $$\\x\cdot\\x=(1-1+0+\cdots)(1-1+0+\cdots)=0\ ,$$ but $\\x\ne\\0$.